I'm following a lecture in the field of stochastic processes in neuroscience where we are working with objects called time-dependent or non-homogeneous renewal processes. However we never defined these processes. Does anybody have any intuition or definition what these processes are?
We defined renewal processes via i.i.d. waiting times i.e. let $(S_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of rv's and define $T_{n+1}:=\sum_{k=1}^{n+1}S_k$. Then $$N(C):=\sum_{n=0}^\infty \mathbb{1}_C(T_n)$$ defines a stochastic point process called renewal process.
Now we never defined the notion of a time-inhomogeneous renewal process, but the lecturer mentioned that they generalise renewal processes in a similar as a Poisson Process with time dependent intensity would generalise a Poisson processes with constant intensity.